Path 1

# tables for success
succ_tabs <- with(dat1 %>% filter(experiment==1), table(success, trial_type, path))
succ1 <- knitr::kable(
  succ_tabs[, , 1],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = succ1, latex_options = 'hold_position')

Path 2

succ2 <- knitr::kable(
  succ_tabs[, , 2],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = succ2, latex_options = 'hold_position')

Path 3

succ3 <- knitr::kable(
  succ_tabs[, , 3],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = succ3, latex_options = 'hold_position')

Success Plotting

Below is a plot demonstrating the success and failures for with and without the belt over each of the three paths. A point near 1.00 indicates a success and a point near 0.00 indicates a failure. Each of the three plots are separated by path type. On the left hand side of each plot, the (red) points denote runs without the belt, and on the right and side of each plot, the (blue) points denote runs with the belt.

# Success Plotting
ggplot(data = dat1 %>% filter(experiment==1), 
       aes(x=trial_type, y = success.bin, color = trial_type), size = 3)+
  geom_jitter(width = 0.5, height=0.001)+
  facet_grid(~path)

Alternative Plotting

The figure below contains simple line plots for each of the subjects. There are a total of three plots for each subject, representing the three paths taken. In each line plot, the point on the left denotes whether the subject was successful (around 1.00) or was a failure (around 0.00) without the belt, and the point on the right denotes the same but with the belt. Hence, a horizontal line at 1 indicates the subject was successful both with and without the belt, whereas a horizontal line at 0 indicates the subject failed both with and without the belt. A line with a positive slope indicates the subject failed without the belt and succeeded with the belt, which is what we would like to see. A line with a negative slope indicates the subject succeeded without the belt and failed with the belt, which is not necessarily what we want to see.

# Alternative Plotting
ggplot(data = dat1 %>% filter(experiment==1, participant_number %in% c(4, 6, 7, 8, 9, 10)), 
       aes(x=trial_type, y = success.bin, color = trial_type), size = 3)+
  geom_point()+
  geom_line(aes(group = interaction(participant_number)))+
  facet_wrap(participant_number~path, ncol = 3)

ggplot(data = dat1 %>% filter(experiment==1, participant_number %in% c(11, 12, 13, 14, 15, 17)), 
       aes(x=trial_type, y = success.bin, color = trial_type), size = 3)+
  geom_point()+
  geom_line(aes(group = interaction(participant_number)))+
  facet_wrap(participant_number~path, ncol = 3)

ggplot(data = dat1 %>% filter(experiment==1, participant_number %in% c(18, 19, 20, 21, 23, 24)), 
       aes(x=trial_type, y = success.bin, color = trial_type), size = 3)+
  geom_point()+
  geom_line(aes(group = interaction(participant_number)))+
  facet_wrap(participant_number~path, ncol = 3)

Random Effects

We expect the random effect terms for each of the participants to be approximately normally distributed, and centered at 0.

Path Distance Traveled

Recall, we expected all of the tests for means to be insignificant and the tests for variances to be significant based on previous boxplots. We can see that the boxplots containing the paired data show something different. Below is a plot with all paths included.

# package for ggplots
# library(ggplot2)

# look at change in path distance with/without belt
# experiment 1 
ggplot(data = dat2 %>% filter(experiment==1), aes(x = trial_type,
                                                  y = path_distance_meters))+
  geom_boxplot()+
  geom_point(aes(color = path), size = 3)+
  geom_line(aes(color = path, group = path))

Plots separated by path: Path 1, then Path 2, then Path 3.

# path distance 
ggplot(data = dat2 %>% filter(experiment==1, path == "E1P1"), aes(x = trial_type, 
                                                  y = path_distance_meters))+
  geom_boxplot()+
  geom_point(size = 3)+
  geom_line()

ggplot(data = dat2 %>% filter(experiment==1, path == "E1P2"), aes(x = trial_type, 
                                                  y = path_distance_meters))+
  geom_boxplot()+
  geom_point(size = 3)+
  geom_line()

ggplot(data = dat2 %>% filter(experiment==1, path == "E1P3"), aes(x = trial_type, 
                                                  y = path_distance_meters))+
  geom_boxplot()+
  geom_point(size = 3)+
  geom_line()

Plot separated by participant.

ggplot(data = succ_dat %>% filter(experiment==1), aes(x = trial_type, 
                        y = path_distance_meters))+
  geom_boxplot()+
  geom_point( aes(color = path), size = 3)+
  geom_line(aes(color = path, group = path))+
  facet_wrap(~participant_number, scales = "free_y")

Plot where boxplots are separated by path and belt vs. no belt.

ggplot(data = succ_dat %>% filter(experiment==1), 
       aes(x = interaction(trial_type,path), 
           y = path_distance_meters))+
  facet_wrap(~experiment)+
  geom_boxplot()+
  geom_jitter(width = 0.1, height = 0, aes(color = path))+
  theme(axis.text.x=element_text(angle = 90))

Plot only separated by belt vs. no belt.

ggplot(data = succ_dat %>% filter(experiment == 1), 
       aes(x = interaction(trial_type), 
           y = path_distance_meters))+
  geom_boxplot()+
  geom_jitter(width = 0.1, height = 0, aes(color = trial_type))+
  theme(axis.text.x=element_text(angle = 90))

Time Spent

All paths in one plot.

# look at change in time spent with/without belt
# experiment 1 
ggplot(data = dat2 %>% filter(experiment==1), aes(x = trial_type, 
                                                  y = time_seconds))+
  geom_boxplot()+
  geom_point(aes(color = path), size = 3)+
  geom_line(aes(color = path, group = path))

Plots separated by path: Path 1, then Path 2, then Path 3.

# Time Spent
ggplot(data = dat2 %>% filter(experiment==1, path == "E1P1"), aes(x = trial_type, 
                                                  y = time_seconds))+
  geom_boxplot()+
  geom_point(size = 3)+
  geom_line()

ggplot(data = dat2 %>% filter(experiment==1, path == "E1P2"), aes(x = trial_type, 
                                                  y = time_seconds))+
  geom_boxplot()+
  geom_point(size = 3)+
  geom_line()

ggplot(data = dat2 %>% filter(experiment==1, path == "E1P3"), aes(x = trial_type, 
                                                  y = time_seconds))+
  geom_boxplot()+
  geom_point(size = 3)+
  geom_line()

Plot separated by participant.

ggplot(data = succ_dat %>% filter(experiment==1), aes(x = trial_type, 
                        y = time_seconds))+
  geom_boxplot()+
  geom_point( aes(color = path), size = 3)+
  geom_line(aes(color = path, group = path))+
  facet_wrap(~participant_number, scales = "free_y")

Plot where boxplots are separated by path and belt vs. no belt.

ggplot(data = succ_dat %>% filter(experiment==1), 
       aes(x = interaction(trial_type,path), 
           y = time_seconds))+
  facet_wrap(~experiment)+
  geom_boxplot()+
  geom_jitter(width = 0.1, height = 0, aes(color = path))+
  theme(axis.text.x=element_text(angle = 90))

Plot only separated by belt vs. no belt.

ggplot(data = succ_dat %>% filter(experiment == 1), 
       aes(x = interaction(trial_type), 
           y = time_seconds))+
  geom_boxplot()+
  geom_jitter(width = 0.1, height = 0, aes(color = trial_type))+
  theme(axis.text.x=element_text(angle = 90))

Paired T-Tests for Group Means - Path Distance Traveled

The reason for choosing a paired t-test is due to the fact that two populations are dependent as it is the same group of people in the group with the belt and the group without the belt. We are running these tests by conditioning on success and separating by path in each of the two experiments. Since a paired test requires equal sample sizes from the two samples, we can only consider pairs of trials where the individual completed the same path both with and without the belt.

The null hypothesis is that there is no significant difference in group means, and the alternative is that there is a significant difference in the group means. Thus, small p-values (generally p-value < 0.05) imply that we reject the null and may conclude that there is sufficient evidence to conclude there is a significant difference in group means. Otherwise, we are unable to conclude that there is a significant difference between group means.

Results from the paired t-tests are provided in the tables below. From top to bottom, we see the results from path 1, then path 2, and then path 3.

# function to create data frame and eventually table of paired t-test results
tab_ttest <- function(test) {
  data.frame(
    est = as.numeric(test$estimate),
    lower95 = test$conf.int[1],
    upper95 = test$conf.int[2],
    t = as.numeric(test$statistic),
    pval = test$p.value,
    row.names = ""
  )
}

# run the paired t tests for each path in experiment 1 separately
dist_test11 <- t.test(yes_belt_path_dist11, no_belt_path_dist11, paired = TRUE)
dist_test12 <- t.test(yes_belt_path_dist12, no_belt_path_dist12, paired = TRUE)
dist_test13 <- t.test(yes_belt_path_dist13, no_belt_path_dist13, paired = TRUE)

The table below shows the results from the paired t-test for path 1. The p-value for the test is approximately 0.0543, which is larger than 0.05, and thus, indicates that there is not sufficient evidence that there is a difference in group means.

dist11 <- knitr::kable(
  tab_ttest(dist_test11),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = dist11, latex_options = 'hold_position')

The table below shows the results from the paired t-test for path 2. The p-value for the test is approximately 0.1895, which is larger than 0.05, and thus, indicates that there is not sufficient evidence that there is a difference in group means.

dist12 <- knitr::kable(
  tab_ttest(dist_test12),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = dist12, latex_options = 'hold_position')

The table below shows the results from the paired t-test for path 3. The p-value for the test is approximately 0.1015, which is larger than 0.05, and thus, indicates that there is not sufficient evidence that there is a difference in group means.

dist13 <- knitr::kable(
  tab_ttest(dist_test13),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = dist13, latex_options = 'hold_position')

The results for the paired t-tests for path distance traveled over all three paths were insignificant, i.e., we did not reject the null hypothesis. Thus, we may conclude that there is not a significant difference in group means, which is what we desired.

Paired T-Tests for Group Means - Time Spent

# select only the columns we need for the t test on time spent
succ_dat_time <- succ_dat %>% select(time_seconds, trial_type, experiment, path)

# time spent for with the belt for experiment 1
yes_belt_time11 <- as.data.frame(succ_dat_time %>% 
                                   filter(trial_type == "yes_belt", experiment == 1, 
                                          path == "E1P1") %>%
                                   select(time_seconds))[, 1]
yes_belt_time12 <- as.data.frame(succ_dat_time %>% 
                                  filter(trial_type == "yes_belt", experiment == 1, 
                                         path == "E1P2") %>%
                                  select(time_seconds))[, 1]
yes_belt_time13 <- as.data.frame(succ_dat_time %>% 
                                  filter(trial_type == "yes_belt", experiment == 1, 
                                         path == "E1P3") %>%
                                  select(time_seconds))[, 1]

# path distance for without the belt for experiment 1
no_belt_time11 <- as.data.frame(succ_dat_time %>% 
                                  filter(trial_type == "no_belt", experiment == 1, 
                                         path == "E1P1") %>%
                                  select(time_seconds))[, 1]
no_belt_time12 <- as.data.frame(succ_dat_time %>% 
                                 filter(trial_type == "no_belt", experiment == 1, 
                                        path == "E1P2") %>%
                                 select(time_seconds))[, 1]
no_belt_time13 <- as.data.frame(succ_dat_time %>% 
                                 filter(trial_type == "no_belt", experiment == 1, 
                                        path == "E1P3") %>%
                                 select(time_seconds))[, 1]

# run the paired t tests for each path in experiment 1 separately
time_test11 <- t.test(yes_belt_time11, no_belt_time11, paired = TRUE)
time_test12 <- t.test(yes_belt_time12, no_belt_time12, paired = TRUE)
time_test13 <- t.test(yes_belt_time13, no_belt_time13, paired = TRUE)

The table below shows the results from the paired t-test for path 1. The p-value for the test is approximately 0.8288, which is larger than 0.05, and thus, indicates that there is not sufficient evidence that there is a difference in group means.

time11 <- knitr::kable(
  tab_ttest(time_test11),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = time11, latex_options = 'hold_position')

The table below shows the results from the paired t-test for path 2. The p-value for the test is approximately 0.6036, which is larger than 0.05, and thus, indicates that there is not sufficient evidence that there is a difference in group means.

time12 <- knitr::kable(
  tab_ttest(time_test12),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = time12, latex_options = 'hold_position')

The table below shows the results from the paired t-test for path 3. The p-value for the test is approximately 0.3400, which is larger than 0.05, and thus, indicates that there is not sufficient evidence that there is a difference in group means.

time13 <- knitr::kable(
  tab_ttest(time_test13),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = time13, latex_options = 'hold_position')

The results for the paired t-tests for time spent over all three paths were insignificant. Thus, we may conclude that there is not a significant difference in group means, which is what we desired.

Wilcox Based Test for Equality in Group Variances - Path Distance Traveled

To test equality of variances in each scenario, we are using an approach explained in a Wilcox paper. Since the two samples are dependent, we needed to be careful with which test we chose. Wilcox’s approach considers heteroskedasticity in the data, it is straightforward to implement, and is easily interpreted. Once again, this requires equal sized paired samples. We expect that there will be a significant difference in variances for each of the situations.

The null hypothesis essentially states that the variances are equal. If we obtain significant evidence against the null hypothesis, we may conclude that there is a difference in the group variances. Otherwise, we are unable to claim that there is a significant difference in the group variances. Significant means that the p-value is less than the level of significance, which is generally taken to be 0.05.

##############################################################################
# Conditioned on Success: test from Wilcox Paper for difference in variances #
##############################################################################

# the idea here is to show that there is a difference in variances between ...
# ... the groups; we see this in the plots
# since the two samples are dependent, we can't use a typical F test or ...
# ... other common tests used for equality of variances
# we will be using a test proposed by Wilcox which takes into account the ...
# ... heteroskedasticity in the data

# test for path distance
U_path11 <- yes_belt_path_dist11 + no_belt_path_dist11
V_path11 <- yes_belt_path_dist11 - no_belt_path_dist11
U_path12 <- yes_belt_path_dist12 + no_belt_path_dist12
V_path12 <- yes_belt_path_dist12 - no_belt_path_dist12
U_path13 <- yes_belt_path_dist13 + no_belt_path_dist13
V_path13 <- yes_belt_path_dist13 - no_belt_path_dist13

# fit linear model
m_path11 <- lm(U_path11 ~ V_path11)
m_path12 <- lm(U_path12 ~ V_path12)
m_path13 <- lm(U_path13 ~ V_path13)

# package for hccm function
library(car)
# heteroskedasticity corrected covariance
cov11 <- hccm(m_path11)
cov12 <- hccm(m_path12)
cov13 <- hccm(m_path13)
# package for the coeftest function
library(lmtest)
# test for equal variances
vartest11 <- coeftest(m_path11, vcov. = cov11)
vartest12 <- coeftest(m_path12, vcov. = cov12)
vartest13 <- coeftest(m_path13, vcov. = cov13)

The table below shows the results of the Wilcox based approach to test equality of variances for path 1 (path distance traveled). The p-value for the test is approximately 0.000016, which is less than 0.05. Thus, there is sufficient evidence to conclude that there is a difference in group variances.

# results for path 1 (path distance traveled)
vdist11 <- knitr::kable(
  vartest11[1:2, ],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = vdist11, latex_options = 'hold_position')

The table below shows the results of the Wilcox based approach to test equality of variances for path 2 (path distance traveled). The p-value for the test is approximately 0.0339, which is less than 0.05. Thus, there is sufficient evidence to conclude that there is a difference in group variances.

# results for path 2 (path distance traveled)
vdist12 <- knitr::kable(
  vartest12[1:2, ],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = vdist12, latex_options = 'hold_position')

The table below shows the results of the Wilcox based approach to test equality of variances for path 3 (path distance traveled). The p-value for the test is approximately 0.000019, which is less than 0.05. Thus, there is sufficient evidence to conclude that there is a difference in group variances.

# results for path 3 (path distance traveled)
vdist13 <- knitr::kable(
  vartest13[1:2, ],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = vdist13, latex_options = 'hold_position')

In all three tests for equal variances for path distance traveled, we see that there is sufficient evidence indicating that the group variances are significantly different, which is what we desired.

Wilcox Based Test for Equality in Group Variances - Time Spent

# test for time spent
U_time11 <- yes_belt_time11 + no_belt_time11
V_time11 <- yes_belt_time11 - no_belt_time11
U_time12 <- yes_belt_time12 + no_belt_time12
V_time12 <- yes_belt_time12 - no_belt_time12
U_time13 <- yes_belt_time13 + no_belt_time13
V_time13 <- yes_belt_time13 - no_belt_time13

# fit linear model
m_time11 <- lm(U_time11 ~ V_time11)
m_time12 <- lm(U_time12 ~ V_time12)
m_time13 <- lm(U_time13 ~ V_time13)

# heteroskedasticity corrected covariance
cov11_time <- hccm(m_time11)
cov12_time <- hccm(m_time12)
cov13_time <- hccm(m_time13)

# test for equal variances
vartime11 <- coeftest(m_time11, vcov. = cov11_time)
vartime12 <- coeftest(m_time12, vcov. = cov12_time)
vartime13 <- coeftest(m_time13, vcov. = cov13_time)

The table below shows the results of the Wilcox based approach to test equality of variances for path 1 (time spent). The p-value for the test is approximately 0.0049, which is less than 0.05. Thus, there is sufficient evidence to conclude that there is a difference in group variances.

# results for path 1 (time spent)
vtime11 <- knitr::kable(
  vartime11[1:2, ],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = vtime11, latex_options = 'hold_position')

The table below shows the results of the Wilcox based approach to test equality of variances for path 2 (time spent). The p-value for the test is approximately 0.4813, which is greater than 0.05. Thus, there is not sufficient evidence to conclude a difference in group variances.

# results for path 2 (time spent)
vtime12 <- knitr::kable(
  vartime12[1:2, ],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = vtime12, latex_options = 'hold_position')

The table below shows the results of the Wilcox based approach to test equality of variances for path 3 (time spent). The p-value for the test is approximately 0.0770, which is greater than 0.05. Thus, there is not sufficient evidence to conclude a difference in group variances.

# results for path 3 (time spent)
vtime13 <- knitr::kable(
  vartime13[1:2, ],
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = vtime13, latex_options = 'hold_position')

In equal variance tests for path 2 and 3, there is not a significant difference in group variances, which is not what we expected.

Path 1 - Start Bearing

Below is a grid containing two plots denoting the start bearing over all subjects for path 1, both with and without the belt.

library(cowplot) # for grids

# Path 1 - start bearing
# no belt
start_nobelt_p <-
ggplot(data = dat1 %>% filter(path=="E1P1"), 
         aes(x = start_bearing_no_belt, y =1))+
  geom_point()+
  geom_point(aes(x = 346, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

# belt
start_belt_p <-
ggplot(data = dat1 %>% filter(path=="E1P1"), 
         aes(x = start_bearing_with_belt, y =1))+
  geom_point()+
  geom_point(aes(x = 346, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

plot_grid(start_nobelt_p, start_belt_p, ncol = 2, 
          labels = "E1P1")

Path 1 - End Bearing

Below is a grid containing two plots denoting the end bearing over all subjects for path 1, both with and without the belt.

# path 1 - end bearing
# no belt
end_nobelt_p <-
ggplot(data = dat1 %>% filter(path=="E1P1", `trial_type`=="yes_belt"), 
         aes(x = `end_bearing_no_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 179, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

# belt
end_belt_p <-
ggplot(data = dat1 %>% filter(path=="E1P1"), 
         aes(x = `end_bearing_with_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 179, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

plot_grid(end_nobelt_p, end_belt_p, ncol = 2, 
          labels = "E1P1")

Path 2 - Start Bearing

Below is a grid containing two plots denoting the start bearing over all subjects for path 2, both with and without the belt.

# path 2 - start bearing
# no belt
start_nobelt_p <-
ggplot(data = dat1 %>% filter(path=="E1P2"), 
         aes(x = `start_bearing_no_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 143, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

# belt
start_belt_p <-
ggplot(data = dat1 %>% filter(path=="E1P2"), 
         aes(x = `start_bearing_with_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 143, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

plot_grid(start_nobelt_p, start_belt_p, ncol = 2, 
          labels = "E1P2")

Below is a grid containing two plots denoting the end bearing over all subjects for path 1, both with and without the belt.

# path 2 - end bearing
# no belt
end_nobelt_p <-
ggplot(data = dat1 %>% filter(path=="E1P2", `trial_type`=="yes_belt"), 
         aes(x = `end_bearing_no_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 315, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

# belt
end_belt_p <-
ggplot(data = dat1 %>% filter(path=="E1P2"), 
         aes(x = `end_bearing_with_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 315, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

plot_grid(end_nobelt_p, end_belt_p, ncol = 2, 
          labels = "E1P2")

Path 3 - Start Bearing

Below is a grid containing two plots denoting the start bearing over all subjects for path 3, both with and without the belt.

# path 3 - start bearing
# no belt
start_nobelt_p <-
ggplot(data = dat1 %>% filter(path=="E1P3"), 
         aes(x = `start_bearing_no_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 302, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

# belt
start_belt_p <-
ggplot(data = dat1 %>% filter(path=="E1P3"), 
         aes(x = `start_bearing_with_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 302, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

plot_grid(start_nobelt_p, start_belt_p, ncol = 2, 
          labels = "E1P3")

Below is a grid containing two plots denoting the end bearing over all subjects for path 3, both with and without the belt.

# path 3 - end bearing
# no belt
end_nobelt_p <-
ggplot(data = dat1 %>% filter(path=="E1P3", `trial_type`=="yes_belt"), 
         aes(x = `end_bearing_no_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 119, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

# belt
end_belt_p <-
ggplot(data = dat1 %>% filter(path=="E1P3"), 
         aes(x = `end_bearing_with_belt`, y =1))+
  geom_point()+
  geom_point(aes(x = 119, y = 1), size= 3, color = "red")+
  scale_x_continuous(breaks=c(0, 90, 180, 270, 360), limits = c(0, 360))+
  coord_polar(start = 0)+
  theme_bw()

plot_grid(end_nobelt_p, end_belt_p, ncol = 2, 
          labels = "E1P3")

Histogram of Differences of Start Bearing for Path 1 - No Belt vs. Belt

The grid below contains two histograms of the differences in start bearing from true bearing, both with and without the belt, for path 1.

library(stringr)
bearing <- dat1 %>% dplyr::select(participant_number,
                           trial_type, path, 
                           starts_with("start"),
                           starts_with("end"),
                           starts_with("actual"))

bearing_start_no_belt <- bearing %>% 
  filter(!is.na(start_bearing_no_belt), path=="E1P1") %>% data.frame()

bearing_start_belt <- bearing %>% 
  filter(!is.na(`start_bearing_with_belt`), path=="E1P1") %>% data.frame()

# bearing_start_no_belt %>% group_by(participant_number)%>% tally() %>%
#   data.frame()
# 
# bearing_start_belt %>% group_by(participant_number)%>% tally() %>%
#   data.frame()


bearing_start_no_belt$condition <- "no belt"
bearing_start_belt$condition <- "belt"

# # participant 17 is problematic: missing start bearing no belt E2P1
# 
# bearing_start_no_belt <- bearing_start_no_belt %>% 
#                           filter(!participant.number==17)
# bearing_start_belt <- bearing_start_belt %>% 
#                           filter(!participant.number==17)

bearing_start_no_belt$diffs <- 
  180 - abs(abs(bearing_start_no_belt$start_bearing_no_belt - 
                  bearing_start_no_belt$actual_start_bearing) - 180)
h1nb_st <- hist(bearing_start_no_belt$diffs)

sum1nb_st <- summary(bearing_start_no_belt$diffs)

bearing_start_belt$diffs <-   
  180 - abs(abs(bearing_start_belt$start_bearing_with_belt - 
                  bearing_start_belt$actual_start_bearing) - 180)
h1b_st <- hist(bearing_start_belt$diffs)

sum1b_st <- summary(bearing_start_belt$diffs)

# plot_grid(h1nb_st, h1b_st, ncol = 2, 
#           labels = "E1P1")

Histogram of Differences of End Bearing for Path 1 - No Belt vs. Belt

The grid below contains two histograms of the differences in end bearing from true bearing, both with and without the belt, for path 1.

bearing_end_no_belt <- bearing %>% 
  filter(!is.na(`end_bearing_no_belt`), path=="E1P1", `trial_type`=="yes_belt") %>% data.frame()

bearing_end_belt <- bearing %>% 
  filter(!is.na(`end_bearing_with_belt`), path=="E1P1") %>% data.frame()

# bearing_end_no_belt %>% group_by(participant.number)%>% tally() %>%
#   data.frame()
# 
# bearing_end_belt %>% group_by(participant.number)%>% tally() %>%
#   data.frame()

bearing_end_no_belt$condition <- "no belt"
bearing_end_belt$condition <- "belt"

bearing_end_no_belt$diffs <- 
  180 - abs(abs(bearing_end_no_belt$end_bearing_no_belt - 
                  bearing_end_no_belt$actual_end_bearing) - 180)
hist(bearing_end_no_belt$diffs)

sum1nb_end <- summary(bearing_end_no_belt$diffs)

bearing_end_belt$diffs <-   180 - 
  abs(abs(bearing_end_belt$end_bearing_with_belt - 
                  bearing_end_belt$actual_end_bearing) - 180)
hist(bearing_end_belt$diffs)

sum1b_end <- summary(bearing_end_belt$diffs)

Histogram of Differences of Start Bearing for Path 2 - No Belt vs. Belt

The grid below contains two histograms of the differences in start bearing from true bearing, both with and without the belt, for path 2.

# path 2
bearing_start_no_belt2 <- bearing %>% 
  filter(!is.na(start_bearing_no_belt), path=="E1P2") %>% data.frame()

bearing_start_belt2 <- bearing %>% 
  filter(!is.na(`start_bearing_with_belt`), path=="E1P2") %>% data.frame()

# bearing_start_no_belt2 %>% group_by(participant_number)%>% tally() %>%
#   data.frame()
# 
# bearing_start_belt2 %>% group_by(participant_number)%>% tally() %>%
#   data.frame()


bearing_start_no_belt2$condition <- "no belt"
bearing_start_belt2$condition <- "belt"

bearing_start_no_belt2$diffs <- 
  180 - abs(abs(bearing_start_no_belt2$start_bearing_no_belt - 
                  bearing_start_no_belt2$actual_start_bearing) - 180)
h2nb_st <- hist(bearing_start_no_belt2$diffs)

sum2nb_st <- summary(bearing_start_no_belt2$diffs)

bearing_start_belt2$diffs <-   
  180 - abs(abs(bearing_start_belt2$start_bearing_with_belt - 
                  bearing_start_belt2$actual_start_bearing) - 180)
h2b_st <- hist(bearing_start_belt2$diffs)

sum2b_st <- summary(bearing_start_belt2$diffs)

# plot_grid(h2nb_st, h2b_st, ncol = 2, 
#           labels = "E1P2")

Histogram of Differences of End Bearing for Path 2 - No Belt vs. Belt

The grid below contains two histograms of the differences in end bearing from true bearing, both with and without the belt, for path 2.

bearing_end_no_belt2 <- bearing %>% 
  filter(!is.na(`end_bearing_no_belt`), path=="E1P2", `trial_type`=="yes_belt") %>% data.frame()

bearing_end_belt2 <- bearing %>% 
  filter(!is.na(`end_bearing_with_belt`), path=="E1P2") %>% data.frame()

# bearing_end_no_belt %>% group_by(participant.number)%>% tally() %>%
#   data.frame()
# 
# bearing_end_belt %>% group_by(participant.number)%>% tally() %>%
#   data.frame()

bearing_end_no_belt2$condition <- "no belt"
bearing_end_belt2$condition <- "belt"

bearing_end_no_belt2$diffs <- 
  180 - abs(abs(bearing_end_no_belt2$end_bearing_no_belt - 
                  bearing_end_no_belt2$actual_end_bearing) - 180)
hist(bearing_end_no_belt2$diffs)

sum2nb_end <- summary(bearing_end_no_belt2$diffs)

bearing_end_belt2$diffs <-   180 - 
  abs(abs(bearing_end_belt2$end_bearing_with_belt - 
                  bearing_end_belt2$actual_end_bearing) - 180)
hist(bearing_end_belt2$diffs)

sum2b_end <- summary(bearing_end_belt2$diffs)

Histogram of Differences of Start Bearing for Path 3 - No Belt vs. Belt

The grid below contains two histograms of the differences in start bearing from true bearing, both with and without the belt, for path 3.

# path 3
bearing_start_no_belt3 <- bearing %>% 
  filter(!is.na(start_bearing_no_belt), path=="E1P3") %>% data.frame()

bearing_start_belt3 <- bearing %>% 
  filter(!is.na(`start_bearing_with_belt`), path=="E1P3") %>% data.frame()

# bearing_start_no_belt3 %>% group_by(participant_number)%>% tally() %>%
#   data.frame()
# 
# bearing_start_belt3 %>% group_by(participant_number)%>% tally() %>%
#   data.frame()

bearing_start_no_belt3$condition <- "no belt"
bearing_start_belt3$condition <- "belt"

bearing_start_no_belt3$diffs <- 
  180 - abs(abs(bearing_start_no_belt3$start_bearing_no_belt - 
                  bearing_start_no_belt3$actual_start_bearing) - 180)
h3nb_st <- hist(bearing_start_no_belt3$diffs)

sum3nb_st <- summary(bearing_start_no_belt3$diffs)

bearing_start_belt3$diffs <-   
  180 - abs(abs(bearing_start_belt3$start_bearing_with_belt - 
                  bearing_start_belt3$actual_start_bearing) - 180)
h3b_st <- hist(bearing_start_belt3$diffs)

sum3b_st <- summary(bearing_start_belt3$diffs)

# plot_grid(h3nb_st, h3b_st, ncol = 2, 
#           labels = "E1P3")

Histogram of Differences of End Bearing for Path 3 - No Belt vs. Belt

The grid below contains two histograms of the differences in end bearing from true bearing, both with and without the belt, for path 3.

bearing_end_no_belt3 <- bearing %>% 
  filter(!is.na(`end_bearing_no_belt`), path=="E1P3", `trial_type`=="yes_belt") %>% data.frame()

bearing_end_belt3 <- bearing %>% 
  filter(!is.na(`end_bearing_with_belt`), path=="E1P3") %>% data.frame()

# bearing_end_no_belt %>% group_by(participant.number)%>% tally() %>%
#   data.frame()
# 
# bearing_end_belt %>% group_by(participant.number)%>% tally() %>%
#   data.frame()

bearing_end_no_belt3$condition <- "no belt"
bearing_end_belt3$condition <- "belt"

bearing_end_no_belt3$diffs <- 
  180 - abs(abs(bearing_end_no_belt3$end_bearing_no_belt - 
                  bearing_end_no_belt3$actual_end_bearing) - 180)
hist(bearing_end_no_belt3$diffs)

sum3nb_end <- summary(bearing_end_no_belt3$diffs)

bearing_end_belt3$diffs <-   180 - 
  abs(abs(bearing_end_belt3$end_bearing_with_belt - 
                  bearing_end_belt3$actual_end_bearing) - 180)
hist(bearing_end_belt3$diffs)

sum3b_end <- summary(bearing_end_belt3$diffs)

Path 1 - Bearing Results

A summary of the differences of start bearing for path 1, with and with the belt, is shown in the table below.

sum1_st <- rbind(sum1nb_st, sum1b_st)
row.names(sum1_st) <- c("no belt", "belt")
sum1st <- knitr::kable(
  sum1_st,
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = sum1st, latex_options = 'hold_position')

The table below shows the results of the t-test for a difference in group means regarding the start bearing for path 1. Note, we are comparing the mean of the differences from the actual start bearing. The p-value is approximately 0.2382, which is greater than 0.05. Thus, there is no significant difference in means between the two groups.

# prepare for paired t-test
start_comp <- 
  rbind(bearing_start_no_belt[, c("participant_number", "condition", "diffs")],
        bearing_start_belt[, c("participant_number", "condition", "diffs")])

start_comp$condition <- factor(start_comp$condition)

startdiffs1 <- t.test(diffs ~ condition,  data = start_comp, paired = TRUE)

stdiffs1 <- knitr::kable(
  tab_ttest(startdiffs1),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = stdiffs1, latex_options = 'hold_position')

A summary of the differences of end bearing for path 1, with and with the belt, is shown in the table below.

sum1_end <- rbind(sum1nb_end, sum1b_end)
row.names(sum1_end) <- c("no belt", "belt")
sum1end <- knitr::kable(
  sum1_end,
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = sum1end, latex_options = 'hold_position')

The table below shows the results of the t-test for a difference in group means regarding the end bearing for path 1. Note, we are comparing the mean of the differences from the actual end bearing. The p-value is approximately 0.4815, which is greater than 0.05. Thus, there is no significant difference in means between the two groups.

end_comp <- 
  rbind(bearing_end_no_belt[, c("participant_number", "condition", "diffs")],
        bearing_end_belt[, c("participant_number", "condition", "diffs")])

end_comp$condition <- factor(end_comp$condition)

enddiffs1 <- t.test(diffs ~ condition,  data = end_comp, paired = TRUE)

ediffs1 <- knitr::kable(
  tab_ttest(enddiffs1),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = ediffs1, latex_options = 'hold_position')

Path 2 - Bearing Results

A summary of the differences of start bearing for path 2, with and with the belt, is shown in the table below.

sum2_st <- rbind(sum2nb_st, sum2b_st)
row.names(sum2_st) <- c("no belt", "belt")
sum2st <- knitr::kable(
  sum2_st,
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = sum2st, latex_options = 'hold_position')

The table below shows the results of the t-test for a difference in group means regarding the start bearing for path 2. Note, we are comparing the mean of the differences from the actual start bearing. The p-value is approximately 0.1356, which is greater than 0.05. Thus, there is no significant difference in means between the two groups.

start_comp2 <- 
  rbind(bearing_start_no_belt2[, c("participant_number", "condition", "diffs")],
        bearing_start_belt2[, c("participant_number", "condition", "diffs")])

start_comp2$condition <- factor(start_comp2$condition)

startdiffs2 <- t.test(diffs ~ condition,  data = start_comp2, paired = TRUE)

stdiffs2 <- knitr::kable(
  tab_ttest(startdiffs2),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = stdiffs2, latex_options = 'hold_position')

A summary of the differences of end bearing for path 2, with and with the belt, is shown in the table below.

sum2_end <- rbind(sum2nb_end, sum2b_end)
row.names(sum2_end) <- c("no belt", "belt")
sum2end <- knitr::kable(
  sum2_end,
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = sum2end, latex_options = 'hold_position')

The table below shows the results of the t-test for a difference in group means regarding the end bearing for path 2. Note, we are comparing the mean of the differences from the actual end bearing. The p-value is approximately 0.0363, which is less than 0.05. Thus, there is a significant difference in means between the two groups.

end_comp2 <- 
  rbind(bearing_end_no_belt2[, c("participant_number", "condition", "diffs")],
        bearing_end_belt2[, c("participant_number", "condition", "diffs")])

end_comp2$condition <- factor(end_comp2$condition)

enddiffs2 <- t.test(diffs ~ condition,  data = end_comp2, paired = TRUE)

ediffs2 <- knitr::kable(
  tab_ttest(enddiffs2),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = ediffs2, latex_options = 'hold_position')

Path 3 - Bearing Results

A summary of the differences of start bearing for path 2, with and with the belt, is shown in the table below.

sum3_st <- rbind(sum3nb_st, sum3b_st)
row.names(sum3_st) <- c("no belt", "belt")
sum3st <- knitr::kable(
  sum3_st,
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = sum3st, latex_options = 'hold_position')

The table below shows the results of the t-test for a difference in group means regarding the start bearing for path 3. Note, we are comparing the mean of the differences from the actual start bearing. The p-value is approximately 0.0126, which is less than 0.05. Thus, there is a significant difference in means between the two groups.

start_comp3 <- 
  rbind(bearing_start_no_belt3[, c("participant_number", "condition", "diffs")],
        bearing_start_belt3[, c("participant_number", "condition", "diffs")])

start_comp3$condition <- factor(start_comp3$condition)

startdiffs3 <- t.test(diffs ~ condition,  data = start_comp3, paired = TRUE)

stdiffs3 <- knitr::kable(
  tab_ttest(startdiffs3),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = stdiffs3, latex_options = 'hold_position')

A summary of the differences of end bearing for path 3, with and with the belt, is shown in the table below.

sum3_end <- rbind(sum3nb_end, sum3b_end)
row.names(sum3_end) <- c("no belt", "belt")
sum3end <- knitr::kable(
  sum3_end,
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = sum3end, latex_options = 'hold_position')

The table below shows the results of the t-test for a difference in group means regarding the end bearing for path 3. Note, we are comparing the mean of the differences from the actual end bearing. The p-value is approximately 0.0021, which is less than 0.05. Thus, there is a significant difference in means between the two groups.

end_comp3 <- 
  rbind(bearing_end_no_belt3[, c("participant_number", "condition", "diffs")],
        bearing_end_belt3[, c("participant_number", "condition", "diffs")])

end_comp3$condition <- factor(end_comp3$condition)

enddiffs3 <- t.test(diffs ~ condition,  data = end_comp3, paired = TRUE)

ediffs3 <- knitr::kable(
  tab_ttest(enddiffs3),
  format.args = list(scientific = F),
  booktabs = T
)
kable_styling(kable_input = ediffs3, latex_options = 'hold_position')